Local precise large and moderate deviations for sums of independent random variables

Abstract

Let {X, X _k: k ≥ 1} be a sequence of independent and identically distributed random variables with a common distribution F. In this paper, the authors establish some results on the local precise large and moderate deviation probabilities for partial sums ${S_n} = \sum\limits_{i = 1}^n {{X_i}} $, in a unified form in which x may be a random variable of an arbitrary type, which state that under some suitable conditions, for some constants T > 0, a and τ > 1/2 and for every fixed γ > 0, the relation $P\left( {{S_n} - na \in \left( {x,\;x + T]} \right)} \right)\~nF\left( {\left( {x + a,\;x + a + T} \right]} \right)$ holds uniformly for all x ≥ γn ^τ as n→∞, that is, $\mathop {\lim }\limits_{n \to + \infty } \mathop {\sup }\limits_{x \geqslant \gamma {n^\tau }} \left| {\frac{{P\left( {{S_n} - na \in \left( {x,\;x + T} \right]} \right)}}{{nF\left( {\left( {x + a,\;x + a + T} \right]} \right)}} - 1} \right| = 0$. The authors also discuss the case where X has an infinite mean.

Keywords

Local precise moderate deviation / Local precise large deviation / Intermediate regularly varying function / O-regularly varying function

References

[1] Aljancic S., Arandelovic D.. O-regularly varying functions. Publ. Inst. Math. (Beograd), 1977, 22: 5-22

[2] Baltrunas A.. A local limit theorem on one-sided large deviations for dominated-variation distributions. Lithuanian Math. J., 1996, 36: 1-7

[3] Berman S. M.. Sojourns and extremes of a diffusion process on a fixed interval. Adv. in Appl. Probab., 1982, 14: 811-832

[4] Bingham N. H., Goldie C. M., Teugels J. L.. Regular Variation, 1987, Cambridge: Cambridge University Press

[5] Cline D. B. H.. Intermediate regular and variation. Proc. London Math. Soc., 1994, 68: 594-616

[6] Doney, R. A., A large deviation local limit theorem, Math. Proc. Cambridge Philos. Soc., 105, 1989, 575–577.

[7] Doney R. A.. One-sided local large deviation and renwal theorems in the case of infinite mean. Probab. Theory Related Fields, 1997, 107: 45-465

[8] Lin J.. A one-sided large deviation local limits theorem. Statist. Probab. Lett., 2008, 78: 2679-2684

[9] Tang Q.. Insensivity to negative dependence of the symptotic behavior of precise large deviations. Electron. J. Probab., 2006, 11: 107-120

[10] Yang Y., Leipus R., Siaulys J.. Local precise large deviations for sums of random variables with O-regularly varying densities. Statist. Probab. Lett., 2010, 80: 1559-1567